1. Hotelling Models, Mixed Strategy NE
Imagine there are two Ideological candidates, who care *only* about the policy
implemented by the winner.
(a) Assume ui(x) = −x2. What is i’s ideal point?
(b) Assume uj (x) = −(1 − x). What is j’s ideal point?
2. Imagine there are two only-office-motivated candidates in a majoritarian system,
who can choose some platform xi ∈ [0, 1]. Voters’ ideal points are distributed over
[0, 1] but you are given no further details of this distribution. If the candidates tie,
the election is determined by a coin toss. If any voter is indifferent, their vote is
determined by a coin toss. (You may not necessarily need both of these assumptions,
but they are good for conceptualizing the problem.)
(a) Formally represent candidate i’s utility, using the following format:
ui(x) =



u1, Condition
u2, Condition
Your conditions should take the form of a summation ∑ over the voters.
(b) What is the NE of this game?
(c) Does this NE behave differently when the number of voters is even versus odd?
(d) What social science phenomenon does this setup illustrate? What knowledge
can we derive from a seemingly trivial result?